Weakly nonlinear stability analysis of a flow past a neo-Hookean solid at arbitrary Reynolds numbers

Abstract

The finite amplitude stability of a plane Couette flow over a deformable solid medium is analyzed with emphasis on the class of high Reynolds number (Re) modes, referred to as the wall modes, for which the viscous stresses are confined to a thin layer adjacent to the fluid-solid interface with thickness O(Re−1/3) times the channel width in the limit Re⪢1. Here, the Reynolds number is defined in terms of the top plate velocity V and the channel width R. Previous linear stability analyses have shown that the wall modes are unstable for Newtonian flow past a linear viscoelastic solid. In the present study, the analysis is extended to examine the weakly nonlinear stability of these unstable wall modes in order to determine the nature of bifurcation of the transition point to finite amplitude states. To account for the finite strain deformations, the flexible solid medium is described by a neo-Hookean elastic model which is a generalization of the commonly used linear constitutive model. The linear stability analysis provides the critical shear rate Γc and the critical wavenumber in the axial direction αc, where the dimensionless shear rate is defined as Γ=ρV2/G, where ρ is the fluid density and G is the shear modulus of the elastic solid. The critical parameter Γc for the neo-Hookean solid is found to be close to Γc for the linear elastic solid analyzed in the previous studies. The first Landau constant s(1), which is the finite amplitude correction to the linear growth rate, is evaluated in the weakly nonlinear stability analysis using both the numerical technique and the high Re asymptotic analysis. The real part of the Landau constant, sr(1), is negative for the wall mode instability in the limit Re⪢1 for a wide range of dimensionless solid thickness H, indicating that there is a supercritical bifurcation of the wall mode instability. The amplitude of the supercritically bifurcated equilibrium state is derived in the vicinity of the critical point. The equilibrium amplitude, in the form A1e2/(Γ−Γc), is found to scale as Re−1/3 in the limit Re⪢1 and is proportional to H2.3 for H⪢1 in the same limit.